Polarisation of SKT Calabi-Yau $\partial \bar{\partial }$-manifolds by Aeppli classes

Abstract

Given a $\partial \bar{\partial }$-manifold $X$ with trivial canonical bundle and carrying a metric $\omega$ such that $\partial \bar{\partial }\omega =0$, we introduce the concept of small deformations of $X$ polarised by the Aeppli cohomology class $[\omega {]}_A$ of an SKT metric $\omega$. There is a correspondence between the manifolds polarised by $[\omega {]}_A$ in the Kuranishi family of $X$ and the Bott-Chern classes that are primitive in a sense that we define. We also investigate the existence of a primitive element in an arbitrary Bott-Chern primitive class and compare the metrics on the base space of the subfamily of manifolds polarised by $[\omega {]}_A$ within the Kuranishi family.

15 pages